Analytic Solutions for a Functional Differential Equation Related to a Traffic Flow Model

Abstract

We study the existence of analytic solutions of a functional differential equation $(z(s)+\alpha {)}^{2}z\text{'}(s)=\beta (z(s+z(s))-z(s))$ which comes from traffic flow model. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the constant $\lambda$ at resonance, that is, at a root of the unity, but also those $\lambda$ near resonance under the Brjuno condition.